## Triangulation and the Hauptvermutung

**
A triangulation
of a topological space is a homeomorphism to simplicial complex.**

The Hauptvermutung
is the conjecture that any two triangulations of a topological space are combinatorially equivalent.
Ueber die topologischen Invarianten mehrdimensionaler
Mannigfaltigkeiten by
H. Tietze,

Monatshefte fuer Math. und Phys. 19, 1-118 (1908).

English translation (2008) by
John Stillwell.
Beitraege zur Analysis situs by
E.
Steinitz, Sitz.-Ber. Berl. Math. Ges. 7, 29-49 (1908)
Die Topologie der Mannigfaltigkeiten by
H. Kneser,
Jahresb. D.M.V. 34, 1-13 (1926)
Page 152 of
Topologie by
P.
Alexandroff and
H.
Hopf

Springer (1935)
A proof of the generalized Schoenflies theorem
by M.Brown, Bull. A.M.S. 66, 74--76 (1960)
Two complexes which are homeomorphic but combinatorially distinct
by J.Milnor, Ann. of Math. 74, 575--590 (1961)
Microbundles I.
by J.Milnor, Topology 3, suppl. 1, 53--80 (1964)
Microbundles are fibre bundles
by J.Kister, Ann. of Math. 80, 190--199 (1964)
La classification des immersions
combinatoires by A.Haefliger and V.Poenaru, Pub. Math. I.H.E.S. 23, 75-91 (1964)
Triangulating homotopy equivalences,
Princeton Ph.D. thesis of D.Sullivan (1965)
On the Hauptvermutung for manifolds,
by D.Sullivan, Bull. A.M.S. 73, 598--600 (1967)
The Hauptvermutung Book
(edited by A.R.) K-Monographs in Mathematics 1, Springer (ex-Kluwer) (1996)

Mathematical Reviews review
Zentralblatt review

Individual papers:

On the Hauptvermutung by A.Ranicki (1996)

Generalisations and Applications of Block Bundles
by A.Casson (1967)

Triangulating Homotopy Equivalences and Homeomorphisms
by D.Sullivan (1967)

The Princeton Notes on the Hauptvermutung
by M.Armstrong, G.Cooke and C.Rourke (1968)
Stable homeomorphisms and the annulus conjecture,
by R.Kirby, Ann. of Maths. (2) 89, 575--582 (1969)
On the triangulation of manifolds and
the Hauptvermutung by R.Kirby and L.Siebenmann,
Bull. A.M.S. 75, 742--749 (1968)
Counting topological
manifolds by J.Cheeger and J.Kister, Topology 9. 149--151 (1970)
Topological Manifolds by L.Siebenmann,
Proc. 1970 Nice ICM, Gauthier--Villars, Vol. 2, 133--163 (1971)
Some Recent Results on Topological Manifolds
by R.Schultz, American Mathematical Monthly 78, 941--952 (1971)
Geometric periodicity and the invariants
of manifolds by D.Sullivan, in
*Manifolds--Amsterdam 1970 (Proc. Nuffic Summer School)*

Springer Lecture Notes in Mathematics, Vol. 197, 44--75 (1971)
Deformations of spaces of imbeddings,
by R.Edwards and R.Kirby, Annals of Maths. (2) 93, 63-88 (1971)
Recent Advances in Topological Manifolds, Notes of
Part III Lecture course by Andrew Casson, Cambridge (1971).

LATEX version
The canonical Schoenflies theorem
by D.B.Gauld, Proc. A.M.S.27, 603--612 (1971)
Approximating cellular maps by homeomorphisms
by L.Siebenmann, Topology 11, 271--294 (1972)
The triangulation of 3-manifolds
by A.J.S.Hamilton, Quart. J. Math. Oxford Ser. (2) 27, 63--70 (1974)
Foundational Essays on Topological Manifolds, Smoothings,
and Triangulations by R.Kirby and L.Siebenmann,

Annals of Mathematics Studies 88, Princeton (1976).
MR review.

Maybe I'm doin' it wrong (MP3 file) Randy Newman
song referred to on page 283. Lyrics.
Varietes simpliciales d'homologie et
varietes topologiques metrisables by T.Matumoto, Orsay thesis
(1976).

Triangulation of manifolds,
Proc. Symp. Pure Maths. Vol. 32 Part 2, 3--6, AMS (1978)
Classification of simplicial triangulations
of topological manifolds by D.E.Galewski and R.J.Stern,
Bull. AMS 82, 916--918 (1976)

Simplicial triangulations of
topological manifolds, Proc. Symp. Pure Maths. Vol. 32 Part 2, 7--12, AMS (1978)

The recognition problem: what is a topological manifold?
by J.W.Cannon, Bull. A.M.S. 84, 832--866 (1978)
Hauptvermutung
by H.Samelson, American Mathematical Monthly 85, 567--569 (1978)
A geometric proof of Rochlin's theorem
by M.Freedman and R.Kirby, Proc. Symp. Pure Math. XXXII, Part 2, 85--97, AMS (1978)
Classification of simplicial triangulations of
topological manifolds by D.E.Galewski and R.J.Stern, Ann. of Maths. (2) 111, 1--34 (1980)

The topology of manifolds and cell-like maps
by R.D.Edwards, Proc. 1978 ICM Helsinki, 111--127 (1980)
The characterization of topological manifolds of dimension *n≥5*
by J.W.Cannon, Proc. 1978 ICM Helsinki, 449--454 (1980)
Algebraic
L-theory and topological manifolds
by A.Ranicki, Cambridge Tracts in Mathematics 102, CUP (1992)
Epsilon surgery
by S.Ferry and E.K.Pedersen, Proc. 1993 Oberwolfach conference on the
Novikov conjecture, Vol. 2,

LMS Lecture Notes 227, CUP, 167--226 (1995)

Section 13 is particularly relevant.
Some history of the Hauptvermutng (F.Quinn and A.R.)
Letter to the Editor, Mathematical Intelligencer 19 (4), 5 (1997)
On Siebenmann periodicity
by S.Hutt, Pacific J. Math. 183, 291--303 (1998)
Classical and Modern Topology
by S.P. Novikov (2000). MR review.
"Piecewise linear structures on topological manifolds" by Y. Rudyak (available on the Arxiv).
Fragments of geometric topology from the sixties.
by S. Buoncristiano, Geometry
and Topology Monographs 6 (2003)
High dimensional manifold topology
then and now Slides of 3 lectures by A.Ranicki at Siebenmann
conference, Orsay (2005)
Oberwolfach Mini-Workshop on the
Hauptvermutung for High-Dimensional Manifolds 13-19 August, 2006.
Photo Report
A controlled-topology proof of the product structure
theorem by F.Quinn, e-print GT/061031 (2006)
Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture,
by Ciprian Manolescu. e-print http://arxiv.org/abs/1303.2354 (2013)
Aspherical manifolds that cannot be triangulated
by Mike Davis, Jim Fowler and Jean-Francois Lafont.
e-print http://arxiv.org/abs/1304.3730 (2013).
Non-triangulable manifolds
Video of lecture by Rob Kirby on 24 May 2013, at the Bonn Arbeitstagung.
Triangulation of manifolds by Frank Quinn.
e-print http://arxiv.org/abs/1310.7644 (2014).
**Google
for the Hauptvermutung**